Optimal quinary cyclic codes with three zeros

Cyclic codes are an important subclass of linear codes, they not only have good algebraic structure, but also are easy to be encoded and decoded. At present, researchers have constructed many optimal ternary cyclic codes, but the study on quinary cyclic codes is less developed. In this paper, by analyzing the solutions of certain equations over \(\mathbb {F}_{5^m}\) ${\mathbb{F}}_{5^m}$ , we construct some optimal quinary cyclic codes with three zeros and with parameters \([5^m-1, 5^m-2-2m, 4]\) $[5^m-1,5^m-2-2m,4]$ , \([5^m-1, 5^m-2-\frac{3m}{2}, 4]\) $[5^m-1,5^m-2-\frac{3m}{2},4]$ . Moreover, the weight distributions of two classes of their duals are also provided.